Marcel Granollers and Pedro Martinez beat Tomas Machac and Jakub Mensik 7-6(8), 7-6(8) to seal Spain's passage to the last four, where David Ferrer's team will face either Argentina or Germany in the last four on Saturday.
Jaume Munar set Spain up to win the tie by beating Jiri Lehecka 6-3, 6-4 and levelling up the match after Pablo Carreno Busta was beaten by world number 19 Mensik 7-5, 6-4.
No Alcaraz, no worries
Spain's biggest star, Carlos Alcaraz, pulled out of the tournament on Tuesday with a hamstring injury suffered during his defeat to Jannik Sinner in the title match of the ATP Finals on Sunday.
The six-time Grand Slam champion said in the aftermath of that defeat that his hamstring hadn't affected him, even though it was strapped up during a medical timeout.
World number one Alcaraz is one of three top-10 players to have dropped out of the Davis Cup finals in northern Italy, where fans have also been denied the chance to see Sinner and Lorenzo Musetti play for the Azzurri.
The highest-ranked player at the Davis Cup finals is former Olympic champion and world number three Alexander Zverev, who will play for Germany against Argentina later on Thursday.